the radical rational…

in search of innovative ideas with a well-balanced approach for the math classroom

Gallery Walk #ppschat Challenge

A common theme in many chapters of Powerful Problem Solving is Gallery Walks. Several techniques are offered throughout the book, but the common goal is to allow students to view their classmates’ approaches to problems.

One of my faults with online book chats is lack of follow-through. I can sometimes use an extra nudge of accountability. There are often so many great ideas and strategies in the books we are chatting that I get overwhelmed and not sure where to begin. Advice: pick 1 thing. Try it. Reflect. Revise. Try it again.

So here is my attempt at a gallery walk. I simply cut apart a pre-assessment for a Formative Assessment Lesson and each pair of students taped it to a large sticky note, discussed and responded. I was confident in many of the questions, but my goal was to identify the few some students were still struggling to understand completely, mostly questions involving transformations.


1. The large majority are fine with creating a possible equation, given the x-intercepts.


2. Initially these students tried -6, -4 and 2 as their intercepts. I asked them to graph their equation then reread the instructions. Oh. They had read write an equation, looked at the graph for possible intercepts and failed to read the y-intercept of (0, -6). One quickly stated the connection between y-intercept and factored terms and was able to adjust their response with ease. I believe it happens often to see a graph skim question and think we know what we’re supposed to do, only to realize skimming sometimes results in miseed information.


3. Within the lesson, many students quickly realized when a factor was squared it resulted in a “double root” and the graph would not actually pass through the x-axis at that point.

The 4 transformations seemed to causes the most disagreements. These were the ones we discussed folowing our gallery walk. However, it was during the gallery walk most students were able to adjust their thinking.


4.i. Listening to students as they were at the poster helped me realize there was not a solid understanding of the reflection across x-axis and maybe we needed to revisit. Possibly, they are confusing with across y-axis?


ii. A few students disagreed initially, but the convo I overheard was addressing that changing the x-intercepts was not sufficient, they looked at the graphs, then said, the functions needs to be decreasing at the begiining, that’s why you have negative coefficient.


iii. Horizontal translations always seem to trick students up. One disagreement actually stated ‘they subtracted and did not add.” Of course, we definitely followed up with this one.


iv. This pair of students argued over which one was right. The expaned version or factored form. Simple, graph the new equations and compare to see which one translates the original up 3 units.


A1 & A2 I believe they’ve got this one.


B1 & B2 some confusion here due to the extra vertical line in the graphic. This student was also interchanging graph & equation in their statement.

I thought the gallery walk was a good task to overview some common misconceotions. It was not intimidating, students were able to communicate their ideas, compare their own thinking to others. I truly tried to stand back and listen. They were on task, checking each other’s work. Each station allowed them to focus on one idea at a time. They were talking math. Most misconceptions were addressed through their discussions or written comments.

Having a moment to debrief the following day highlighted the big ideas students had addressed the previously and reinforced the corrections they had made. This was so much more valuable than me standing in front of the room telling them which mistakes to watch for. Their quick reflection writes revealed majority have a better ability to transform the functions, which was my initial goal for the gallery walk. A few still have minor misgivings that can be handled on an individual basis.


Number Talks 12 x 13

I was cleaning up some files this evening and ran across these snapshots from a Number Talk in class a while back. 

Are all of them the most efficient?  For me, no.  However, if it made sense to that student at that particular moment, then it was most efficient for them.  I appreciate the various ways they consider “building” the product.

  I wanted to see their thinking on 2 digit multiplication to link back our unit on polynomial operations.








Yes, they all had a calculator available, but my question was, how do you know?  As I learned from Steve Leinwand, “Convince me.”

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KCM 2014 Links

Fabulous, Friendly, Fun

Fluency Forward ~ KCM 2014


Fabulous Starters

*Estimation 180 – Mr. Stadel                   

*Counting Circles – Sadie Estrella

Number Talks –

Jo Boaler                          Stanford Course How to Learn Math

Convince me! – Steve Leinwand

Inside Mathematics

Fawn Nguyen (math talks)

*Odd One Out – Malcolm Swan  


Encourage a Culture of Listening

Powerful Problem Solving, Max Ray


Student Reflection

*Color with Purpose, Interactive Notebooks


1. I use to think…

    But now, I know…

2. HW

        Which 2 problems were most alike?  Explain.

        Which 2 problems were most different?  Explain.

       A particular problem that I struggled with…

3.  Practice / Activity

      Which problems were easiest for me?

      Which problems were most difficult for me?

      Watch for…..

4.  Wrong Answer Analysis, Stiggins

5.  2-Minute Assessment Grid

6. Chalk Talk, Making Thinking Visible

Friendly Interactions with Math

*Open Questions – More Good Questions, Marian Small & Amy Lin

Task #1  Slope is 3/5

Task #2  (0,2) and (5, 0)


*Staircases & Steepness – Fawn Nguyen


*Triangle Centers – Geogebra

Notice / Wonder(The Math Forum)

Amusement Park Placement


* Sol LeWitt (Art Structure) Notice/Wonder (Max Ray/Annie Fetter)


Open Sorts 


Representing Polynomials FAL

*Midpoint Miracle (geogebra)



Shadows (Zoom out) – Kentucky Vietnam Memorial


Fun Opportunities

Graphing Stories –

Formative Assessment Lessons from MARS  

Tom distance/time graphs

               Everyday Situations with Functions

Speed Dating Function of Time Blog


Dice Equations of Lines with some Novelty


Blocks Polynomial Station Activities


Hole Punch Game


Ghosts in the Graveyard Math Tales from the Spring Blogspot


Math Madness Music – Bob Garvey

YouTube Parodies – Westerville South High School Gettin’ Triggy with It, Quad Solve


Engagement Wheel
David Sladkey
Reflections from a High School Math Teacher

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32-12 my opinions

I usually don’t post on things as ridiculous as this.  The comments and posts made me cringe. These attitudes are from people who are uneducated about common core…  It makes me sad to think so many are mis-educated and truly believe this is what CCSS is all about.


CCSS is not about making math more difficult.

I agree this example looks longer than most of us learned the traditional way but CCSS is about allowing students to develop number sense.  IF a student solved a problem this “longer way” -I agree it is not how I would have approached it, but is it incorrect? Is their thinking wrong?  I would like to have a conversation to really hear/see their thinking.  It seems they started at 12 and counted up to 32.  For a student who struggles with subtraction, yet excels in addition, I think this is a perfectly legit approach.

I never remember being allowed to explore different strategies but told how to do the problem and what to think about it.  When students are required to do it “the teacher’s way” many do not think/process the same way, they get frustrated, feel like a failure, hence the reason so many dispise math nowadays.

At some point in my career, I complained I taught something but don’t know why students didn’t get it. So, I retaught it, the same way I did the first time, just more examples and spoke more slowly and expected different results.  Sheesh.

I have complained that “they knew it” on the unit test yet not on a cumulative exam at the end of the year.  Did I spiral review throughout the year?  Did I teach isolated skills?  Did I let them approach it a way that made sense to them?  Did I allow them to work, sharing their strategies with classmates?  Seriously, if you are very traditional in your teaching, watch a struggling student trying to use a procedure, they never really understood, to solve a problem.

I am not saying get rid of good instruction but listen to your students.  If they don’t understand your method/procedure, let them make sense of it in their own way.  Number talks are an amazing way to start listening to student thinking. has some nice examples to consider.

IF you are posting statements to #boycottcommoncore please learn more.  Find out what its really about.


Powerful Problem Solving Winter/Spring Chat #ppschat

Here are links to the Storify for our chats:

Chapter 1 1/22/14 Introduction

Chapter 2 1/29/14  Communication & Community

Chapter 3 2/5/14  Learning Through Listening

Chapter 4 2/12/14 Noticing and Wondering

Chapter 5 2/19/14 Changing Representation:  Seeing the Big Picture

Chapter 6 2/26/14 Engaging Students’ Number Sense with Guessing

Chapter 7  3/5/14 Getting Organized

Chapter 8 3/12/14  Generalizing, Abstracting, and Modeling

Oops!  I somehow waited too long – having trouble rounding up these tweets for Chapter 8!

Chapter 9 3/19/14  Looking for Structure

Chapter 10 3/26/14  The Problem-Solving Process & Metacognition

Chapter 11 4/2/14  Reflecting, Revising, Justifying, and Extending

Chapter 12 4/9/14


My Win that Makes Me Sad

Little things are big wins for me.   A student from a semester class in the fall stops by to say hello often.  The other day A asks if I have any math problems they could work on during their new semester class… 1) they miss doing math everyday 2) they are bored.  I pull out a copy of MT Calendar Problems from last year with the the key and hand to A.

I know A is working on them because A returns to my classroom with a questions because they were having trouble making sense of particular problems…math they haven’t learned yet.  A couple of questions on my part (because I’m the one who answers questions with questions) and A was moving forward again.

When a student responds to “Did you enjoy your break?” with, “Not really.  School is boring, but less boring than home.”  Are we giving them a reason to be here? 

Don’t take this the wrong way…we have many wonderful teachers, most work very, very hard.  But, it makes me sad that I sometimes see how boring I am on my students’ faces.  I walk down the hall peering through doors and seeing students unengaged.  Highly intellectual beings that are not thinking because we are not letting them think.  We are boring them. No, its not like this all day, everyday.  But any moment , in my opinion, they are not engaged is wasted time, a missed opportunity.  We may be working hard, but rather than asking them to think, are we telling them here is what you need (I want you) to know, don’t think about it, just know it (aka memorize for the test and then forget it)?

And they do, some of them anyway.

I don’t have the research but my guess is 30% get the game of school. They can do what is asked with little to no effort. Yes they are “successful” in their academics, complete all of the assignments, get all the right grades, but have we really asked them /given them opportunities to think deeply?

Another 30% want to play the game of school, are very intellectual but their processing is way different than ours.  They try, they want to please us and their parents by hard work but they don’t get it enough to actually retain it beyond the test.  With some adjustments in our approach, we could really move these students forward.

Of the remaining 30%, this is the group we could have the biggest impact with but stepping a little further outside our comfort zone.  Actually, for some, I guess it would be way out of their comfort zone to connect-really connect with these students.

My guess is several are distracted by outside factors beyond our reach.  They lost what small connections they had with educators long ago.  At some point they began to feel disconnected and realized they were being pushed along.  It could have been an undetected learning disability.  Maybe they didn’t want anyone to know they didn’t “get it” so they just stopped trying, and we labeled them lazy.  Yep, there was that one teacher who made some long strides, but the folllowing year, noone kept that same level of interest, so they slid back into their isolation. Once, a student wrote on an evaluation “I appreciate what you’re doing, but what’s the use, if it will go right back the old way when I’m no longer in your class.”

I whole heartedly believe some of our most intellectual students fall into this group.  They don’t buy in to our definition of education (eocs, standardized testing).  Look at what they’re reading… I mean seriously, if a student is not capable or lazy, would they be reading novel after novel, especially things I find over my head? Or books about the most influential physicists of the century?  Have we actually taken the time to appreciate the beauty and details in their notebook of doodles? 

Take a moment to have a real conversation with them.
See beyond what you want them to do… 
They’re a person. 
Somebody’s child, grandchild, niece or nephew, friend. 
Wanting to learn. 

Let them know you see them. 
Let them know you miss them when they’re not in your classroom. 
Let them know they matter.



Let them know you believe in them.
And don’t be fake.
They know fake.


No jumping in, silent/listening, no repeating & my win for the day #ppschat

Our #ppschats the past few weeks have brought some a-has and good reminders for me.  Here are a few adjustments I am trying to mindful of:

1. When students are working in small groups, I have often jumped in to their conversation when I heard them going in the wrong direction.  Of course, my intentions may have been to redirect them. 
Needed adjustment:  Be silent and listen.  Giving them space to muddle through their own thinking without jumping in and telling them what I think they should think.  A key for me may be to keep my tablet, clipboard, post-its to jot down notes of their conversation.  Key points to reference back to later.  Jot down questions I might like to ask.  Not sure when to ask /share, maybe as a quick revisit before the end of class?

2. In my efforts to “value” what a student shares, I often find myself repeating. Afterall, those softspoken students need to be heard, so I repeat it so classmates across the room “hear” them.  Oh, no.

Needed adjustment: Ask student to speak up so others can hear.  If they are intimidated, offer an encouraging word, let them know you like it, find it interesting or you want others to hear it.  When I repeat, I am causing others to not listen because they know I’m going to repeat.  Oh, my.  Guilty.  Who knew?  What are ways you create a listening community of students? 

3.  I may ask for volunteers and the same 8 people are sharing.  Spread the love.

Needed adjustment:  I tried to be very purposeful in sharing this week.  In geometry, I picked a problem several seemed to have trouble with.  I structured the task with Know: what information is given, Notice: what do I notice about the diagram? Other information I can use to move me further?  Wonder: What other measures can I determine? How can I justify my reasoning? 

Students took couple of minutes individually, to jot down a couple of things in each bullet, then in their groups of 3 to discuss.  I asked each group to pick 1 thing they felt was important to share.  Yep.  Good ol’ Think-Pair-Share.  As I went around to groups, I arrived at one who said…they already shared ours, so I used the suggestion from PPS to +1 on the board.  It seemed that others were really listening to what was being said.

A win in class today as I gave students a diagram with no questions.  They noticed/wondered and it was a statement from a student that 2 chords were congruent.  When I asked them to convinece me…their statement was quite fuzzy ending with “it just seems like they would be.” I challenged the others to prove or dispute the statement… 

“Oh yea” high fives, “we got it!” & “you’re genius!”  Students celebrating something they hadn’t seen before. 

What was even better, the way they justified their reasoning…all different, not one that I had seen myself.  And that is why I don’t care for the answer key as the answer key. Students sharing their strategy and each confirming the others.  Hearing them say ‘that’s cool’ to another student’s strategy.  Engaged while looking at a different approach.  I truly feel the take aways from a single problem approached this way is valuable.  It was a productive day.

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Pomegranate & Cream Cheese Crepes (non-math post)

Sunday mornings are the one day I cook breakfast-no poptarts, toast & jelly, bowls of cereal.  A few weeks ago, my husband saw a commercial with waffles, cream cheese and pomegranate.  Thought I would try something similar today.  A quick search on Pinterest resulted in this recipe for Cream Cheese Crepes.  Thouh my pic isn’t nearly as beautiful as the one in the link, for my first attempt, these were a hit!


Filling…blend until smooth with a hand mixer:
8 oz cream cheese
1/4 cup confectioner’s sugar
2 TBsp butter
1 tsp vanilla

1 1/3 cup milk
4 eggs, slihtly beaten
1 tsp vanilla
2 TBsp vegetable oil
1 cup flour
2 TBsp white sugar
1/4 tsp salt

Blend until smooth.
Heat 10 inch skillet, spray.  Using ladel, spoon ~3Tbsp. into skillet. Swirl until bottom of pan is coated. When bubby (1-2 minutes), flip, cook an additional minute. Remove from skillet.  Spread with cream cheese filling, sprinkle with pomegranate or favorite fruit or jam.  Roll up.  Serve warm. 

Who knew crepes were that easy?!?

This recipe make 10-12 crepes.

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Complex Numbers and Speed Dating

About 3 years ago, I ran across this post from Kate Nowak called Speed Dating.  I have used it or a version many times.  Students do love it and often laugh about going home and telling their parents they speed dated in math class today!

She suggested placing desks so students are facing one another like this:


I used 2 colors of markers (one purple, one blue).  Each card contained one problem involving arithmetic, simplyfing complex number expressions, conjugates, graphing in complex plane and finding the modulus. I used the other color marker to create a parallel set of problems.   Each student in one row received a problem from the blue set and the other row from purples.  This enabled every student to experience at least one of each type probelm.


Students solved their problem, I confirmed or asked questions to clarify their work.  They became the expert of that problem.  Every person they “dated” would solve their problem.  As Kate’s post mentioned, you can easily assign problems to specific students to differentiate levels if needed.

Pairs would exchange cards, working the given problem.  The expert would check the other’s work, confirming or asking questions to help them correct any mistakes.  They would get their problem card back.  Front row rotates to next person.  Repeat.  By the end of the round, each student has practiced a variety of problems with immediate feedback as needed.

Exit slips revealed mistakes due to not paying attention to a given operation; a few need reminders of i^2 becoming -1 as one more step to simplify further; several still not comfortable with using conjugate to simplify rational expression of complex numbers.  Students enjoy working with others, an opportunity to be out of their seats.  Its a chance for them to ask questions of their peers in a smaller setting as opposed to whole-class.

Here is another idea…the Placemat Activity @cheesemonkeysf  incorporated to practice arithmetic with complex numbers.  Its amazing how quickly one forgets great little spins to use in the classroom.  Thank goodness for the open sharing of MTBoS to remind me!


Forming Quadratics lesson from MAP

I have planned to share this lesson for several weeks but time has gotten away.  My students were not where they needed to be with quadratics, so I pulled together some tried and true tasks-framing quadratics, Wylie Coyote, et al and a new one from Mathematics Assessment Project called Forming Quadratics. You can download lesson, domino cards and assessment in that link.


No big surprises on the pre-assessment, but I did use it to place students in pairs based on similar thinking/reponses. There are 4 equations students are asked to match to 4 graphs and explain their matches.

I like this lesson for a lot of reasons. Discussion of how different forms give us different information. Allows students to seek key features from graphs, connecting them to parts of different but equivalent forms of equations. Students work in their pair but also must visit other groups to confirm/dispute their responses. The lesson outlines its goals:


This lesson should be used after students are familiar working with different forms of quadratics. This is not an intro lesson, but one I see being successful about 2/3 way through unit or as a follow-up/review activity. They will encounter standard, factored and completed square/vertex forms.

I followed the lesson pretty true to outline, changing only minor things based on my classes. After the whole class intro, pairs worked at matching dominoe-style cards including sets of functions and graphs. I was adament about them taking turns explaining their matches. Some cards had all equation forms, some had only parts. They recorded their matches on a card for the next round.
Following the initial round, one person stayed and another person moved to a different group. In the new groups, they were asked to compare responses, then discuss any differences. This took only a few minutes. Upon returning to original partner, they now had to fill-in missing information on the equations. Again, upon completing their equations, one person stayed and the other traveled to a new partner to compare. Some a-ha’s came about during this part as they maneuvered between the different forms, such as the last term in vertex form does not necessarily correspond to the y-intercept as in standard form. So if and when would they be the same was a nice question for discussion.

As an exit slip this day, students were asked to fill-in front side of this foldable for their INBs.

The following class, I pased back their foldable and gave them a few minutes to respond to my feedback. They received smaller copies of the dominoe cards to cut apart and match inside their foldable. They were asked to write any missing equations, and Color With Purpose different parts of equations and graphs.

I used the same cards and was able to offer some feedback on simple mistakes, but in the future maybe I should have a fresh set and use it as a true formative assessment to see they are able to match new sets & write new equations.

A panel on the trifold was a place to record/review other important info concerning quadratics.


Classes were still somewhat split in the post assessment. 1/3 were right on track, 1/3 had trouble with writing the equations, 1/3 seemed almost clueless- I was like “what happened?” They couldn’t correctly identify key points from the graphs. Before passing back their work, I asked what made it difficult? Even if their matches were correct, they failed to give correct coordinates of key points. Their responses all very similar “there were no values on the graphs.” I passed them back and allowed them to talk over feedack with nearby students. After speaking with them individually, I was convinced all but a couple were now moving in the right direction.

Hmm. Maybe next go, I should scaffold the assessment. Part I, very similar to their practice, including labels on graphs. Part II, similar to current assessment with no labels on graphs. Part III writing equations for given information or identified points from graphs.

All in all, I was satisfied with the discussions students were having; How they had to explain their reasoning for matches made. I see these conversations prooving valuable as we continue in our next unit on other polynomial functions.

A post of the same lesson from Ms. Rudolph.



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